Gravitational lensing: numerical simulations with a hierarchical tree code

The mathematical formulation of gravitational lensing — the lens equation — is a very simple mapping R2→R2, between the lens (or sky) plane and the source plane. This approximation assumes that all the deflecting matter is in one plane. In this case the deflection angle α is just the sum over all mass elements in the lens plane. For certain problems — like the determination of the magnification of sources over a large number of source positions (up to 1010) for very many lenses (up to 106) — straightforward techniques for the determination of the deflection angle are far too slow. We implemented an algorithm that includes a two-dimensional tree-code plus a multipole expansion in order to make such microlensing simulations “inexpensive”. Subsequently we modified this algorithm such that it could be applied to a three-dimensional mass distribution that fills the universe (approximated by many lens planes), in order to determine the imaging properties of cosmological lens simulations. Here we describe the techniques and the numerical methods, and we mention a few astrophysical results obtained with these methods.